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A186757 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing cycles of length >=2 (0<=k<= n/2). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < b(3) < ... .

%I A186757 #17 Dec 01 2021 12:02:32
%S A186757 1,1,1,1,2,4,10,11,3,59,36,25,363,212,130,15,2491,1688,651,210,19661,
%T A186757 14317,4487,1750,105,176536,129076,42435,12628,2205,1767540,1277159,
%U A186757 451626,104755,26775,945,19460671,13974236,5068723,1120570,264880,27720
%N A186757 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k increasing cycles of length >=2 (0<=k<= n/2). A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < b(3) < ... .
%C A186757 Row n contains 1 + floor(n/2) entries.
%C A186757 Sum of entries in row n is n!.
%C A186757 T(n,0) = A186758(n).
%C A186757 Sum_{k>=0} k*T(n,k) = A056542(n).
%H A186757 Alois P. Heinz, <a href="/A186757/b186757.txt">Rows n = 0..200, flattened</a>
%F A186757 E.g.f.: G(t,z) = exp((t-1)(exp(z)-1-z))/(1-z).
%F A186757 The 4-variate e.g.f. H(u,v,w,z) of the permutations of {1,2,...,n} with respect to size (marked by z), number of fixed points (marked by u), number of increasing cycles of length >=2 (marked by v), and number of nonincreasing cycles (marked by w) is given by H(u,v,w,z)=exp(uz+v(exp(z)-1-z)+w(1-exp(z))/(1-z)^w. Remark: the nonincreasing cycles are necessarily of length >=3. We have: G(t,z)=H(1,t,1,z).
%e A186757 T(3,0)=2 because we have (1)(2)(3) and (132).
%e A186757 T(4,2)=3 because we have (13)(24), (12)(34), and (14)(23).
%e A186757 Triangle starts:
%e A186757     1;
%e A186757     1;
%e A186757     1,   1;
%e A186757     2,   4;
%e A186757    10,  11,   3;
%e A186757    59,  36,  25;
%e A186757   363, 212, 130, 15;
%p A186757 b:= proc(n) option remember; expand(
%p A186757       `if`(n=0, 1, add(b(n-i)*binomial(n-1, i-1)*
%p A186757       `if`(i>1, (x+(i-1)!-1), 1), i=1..n)))
%p A186757     end:
%p A186757 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186757 seq(T(n), n=0..12);  # _Alois P. Heinz_, Mar 19 2017
%t A186757 b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n-i]*Binomial[n-1, i-1]*If[i > 1, (x + (i - 1)! - 1), 1], {i, 1, n}]]];
%t A186757 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];
%t A186757 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, May 03 2017, after _Alois P. Heinz_ *)
%Y A186757 Cf. A056542, A186754, A186755, A186756, A186758, A186759, A186760.
%K A186757 nonn,tabf
%O A186757 0,5
%A A186757 _Emeric Deutsch_, Feb 26 2011